The progress of Industry 4.0 and the advancement of robotic design are revealing a significant gap in the capabilities of current manufacturing techniques and the selection of materials that are available in electronics. Present-day electrical systems largely rely on metals, but there is a driving need to develop new electrically conductive objects with a wide range of material properties, including expanded flexibility and softness, and with increasingly complex geometries. Electrically conductive composites can replace traditional metal-based systems. In particular, thermoplastic composites become electrically conductive with the incorporation of conductive fillers or polymers, while retaining to a large extent the processability of the thermoplastic matrix. This is where fused filament fabrication (FFF), an additive manufacturing (AM) technique capable of processing a variety of thermoplastic-matrix feedstock materials, can be leveraged to create electrically conductive objects with new functionalities. While there is an increasing number of publications describing the FFF of electrical objects such as sensors and circuits, there is no comprehensive review outlining the functioning mechanisms, drawbacks and advantages of FFF as applied to conductive materials. The present review fills this lacuna by offering a critical analysis of the specific challenges and solutions to promote FFF of electrically conductive polymers and composites.